Polaris runs from east to west in a straight line. The reason why it appears stationary is because it's situated only 0.5 degrees south of the northern Foundation and makes a flat light wave as the picture above shows.

Polaris can be observed anywhere north of the equator (with the exception of small areas), but as for the South, Polaris is below the horizon anywhere in the South (except small areas) because the light body of Polaris intersect the Earth's surface along the equator.

The remaining question is, Why is polaris visible throughout the year while other constellations come and go?

Reminder: All stars are projected onto the Firmament by the Celestial Sphere. A little bit more than half of the sphere is always projected onto the Firmament, while the upper part of the sphere is not projected and stars and constellation cannot be seen from Earth when they are not projected. Throughout the year, the sphere makes a revolution around the axis illustrated in the picture above - this is why constellations constantly change and and some constellation disappear from the night sky for a period of time. Now back to Polaris...

Polaris is situated right next to the axis of rotation of the Celestial Sphere and remains within the projected area the entire year.